Testing contractibility in planar rips complexes

The (Vietoris-)Rips complex of a discrete point-set <i>P</i> is an abstract simplicial complex in which a subset of <i>P</i> defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires <i>O</i>(<i>m</i> log <i>n</i>) time to preprocess a set of <i>n</i> points in the plane in which <i>m</i> pairs have distance at most 1; after preprocessing, deciding whether a cycle of <i>k</i> Rips edges is contractible requires <i>O</i>(<i>k</i>) time. We also describe an algorithm to compute the shortest non-contractible cycle in a planar Rips complex in <i>O</i>(<i>n</i><sup>2</sup>log <i>n</i> + <i>mn</i>) time.

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